n/d-prism atop 2n/d-prism
in approx. ASCII-art
x-n/d-o x-n/d-x -- n/d-cupola x-n/d-o x-n/d-x -- n/d-cupola | +- 2n/d-prism +---------- n/d-prism
- general polytopal classes:
- segmentochora
| Acronym | n/d-cupe |
| Name |
n/d-cupola prism, n/d-prism atop 2n/d-prism |
| Segmentochoron display |
|
| Circumradius | sqrt[(5+2 cos(π d/n)-4 cos2(π d/n))/(6-8 cos2(π d/n))] |
|
Lace city in approx. ASCII-art |
x-n/d-o x-n/d-x -- n/d-cupola x-n/d-o x-n/d-x -- n/d-cupola | +- 2n/d-prism +---------- n/d-prism |
| Face vector | 6n, 13n, 9n+4, 2n+4 |
| Especially | tisdip (n=2,d=1)* tricupe (n=3,d=1) squacupe (n=4,d=1) pecupe (n=5,d=1) stacupe (n=5,d=2)‡ hicupe (n=6,d=1)* |
| Confer |
|
The height formula given below shows that only 2 < n/d < 6 is possible.
* The maximal height would be obtained at n/d = 2 with upright latteral triangles but then degenerate top base (and thus slightly different incidence matrix),
the other extremal value n/d = 6 would generate a height of zero.
‡ For even denominators the outcome would have a Grünbaumian bottom base.
Incidence matrix according to Dynkin symbol
xx-n/d-ox xx&#x (6/5<n/d<6 and n/d<>2) → height = sqrt[1-1/(4 sin2(π d/n))]
(n/d-p || 2n/d-p)
o.-n/d-o. o. | 2n * | 2 1 2 0 0 0 | 1 2 2 1 2 0 0 0 | 1 1 2 1 0
.o-n/d-.o .o | * 4n | 0 0 1 1 1 1 | 0 0 1 1 1 1 1 1 | 0 1 1 1 1
----------------+-------+------------------+--------------------+----------
x. .. .. | 2 0 | 2n * * * * * | 1 1 1 0 0 0 0 0 | 1 1 1 0 0
.. .. x. | 2 0 | * n * * * * | 0 2 0 0 2 0 0 0 | 1 0 2 1 0
oo-n/d-oo oo&#x | 1 1 | * * 4n * * * | 0 0 1 1 1 0 0 0 | 0 1 1 1 0
.x .. .. | 0 2 | * * * 2n * * | 0 0 1 0 0 1 1 0 | 0 1 1 0 1
.. .x .. | 0 2 | * * * * 2n * | 0 0 0 1 0 1 0 1 | 0 1 0 1 1
.. .. .x | 0 2 | * * * * * 2n | 0 0 0 0 1 0 1 1 | 0 0 1 1 1
----------------+-------+------------------+--------------------+----------
x.-n/d-o. .. | n 0 | n 0 0 0 0 0 | 2 * * * * * * * | 1 1 0 0 0
x. .. x. | 4 0 | 2 2 0 0 0 0 | * n * * * * * * | 1 0 1 0 0
xx .. ..&#x | 2 2 | 1 0 2 1 0 0 | * * 2n * * * * * | 0 1 1 0 0
.. ox ..&#x | 1 2 | 0 0 2 0 1 0 | * * * 2n * * * * | 0 1 0 1 0
.. .. xx&#x | 2 2 | 0 1 2 0 0 1 | * * * * 2n * * * | 0 0 1 1 0
.x-n/d-.x .. | 0 2n | 0 0 0 n n 0 | * * * * * 2 * * | 0 1 0 0 1
.x .. .x | 0 4 | 0 0 0 2 0 2 | * * * * * * n * | 0 0 1 0 1
.. .x .x | 0 4 | 0 0 0 0 2 2 | * * * * * * * n | 0 0 0 1 1
----------------+-------+------------------+--------------------+----------
x.-n/d-o. x. ♦ 2n 0 | 2n n 0 0 0 0 | 2 n 0 0 0 0 0 0 | 1 * * * *
xx-n/d-ox ..&#x ♦ n 2n | n 0 2n n n 0 | 1 0 n n 0 1 0 0 | * 2 * * *
xx .. xx&#x ♦ 4 4 | 2 2 4 2 0 2 | 0 1 2 0 2 0 1 0 | * * n * *
.. ox xx&#x ♦ 2 4 | 0 1 4 0 2 2 | 0 0 0 2 2 0 0 1 | * * * n *
.x-n/d-.x .x ♦ 0 4n | 0 0 0 2n 2n 2n | 0 0 0 0 0 2 n n | * * * * 1
xx-n/d-ox&#x || xx-n/d-ox&#x (6/5<n/d<6 and n/d<>2) → height = 1 (n/d-cu || n/d-cu) o.-n/d-o. || .. .. | n * * * | 2 2 1 0 0 0 0 0 0 0 | 1 2 1 2 2 0 0 0 0 0 0 0 | 1 1 2 1 0 0 .o-n/d-.o || .. .. | * 2n * * | 0 1 0 1 1 1 0 0 0 0 | 0 1 1 0 1 1 1 1 0 0 0 0 | 1 0 1 1 1 0 .. .. || o.-n/d-o. | * * n * | 0 0 1 0 0 0 2 2 0 0 | 0 0 0 2 2 0 0 0 1 2 1 0 | 0 1 2 1 0 1 .. .. || .o-n/d-.o | * * * 2n | 0 0 0 0 0 1 0 1 1 1 | 0 0 0 0 1 0 1 1 0 1 1 1 | 0 0 1 1 1 1 -----------------------------+-----------+------------------------+--------------------------+------------ x. .. || .. .. | 2 0 0 0 | n * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0 oo-n/d-oo&#x || .. .. | 1 1 0 0 | * 2n * * * * * * * * | 0 1 1 0 1 0 0 0 0 0 0 0 | 1 0 1 1 0 0 o.-n/d-o. || o.-n/d-o. | 1 0 1 0 | * * n * * * * * * * | 0 0 0 2 2 0 0 0 0 0 0 0 | 0 1 2 1 0 0 .x .. || .. .. | 0 2 0 0 | * * * n * * * * * * | 0 1 0 0 0 1 1 0 0 0 0 0 | 1 0 1 0 1 0 .. .x || .. .. | 0 2 0 0 | * * * * n * * * * * | 0 0 1 0 0 1 0 1 0 0 0 0 | 1 0 0 1 1 0 .o-n/d-.o || .o-n/d-.o | 0 1 0 1 | * * * * * 2n * * * * | 0 0 0 0 1 0 1 1 0 0 0 0 | 0 0 1 1 1 0 .. .. || x. .. | 0 0 2 0 | * * * * * * n * * * | 0 0 0 1 0 0 0 0 1 1 0 0 | 0 1 1 0 0 1 .. .. || oo-n/d-oo&#x | 0 0 1 1 | * * * * * * * 2n * * | 0 0 0 0 1 0 0 0 0 1 1 0 | 0 0 1 1 0 1 .. .. || .x .. | 0 0 0 2 | * * * * * * * * n * | 0 0 0 0 0 0 1 0 0 1 0 1 | 0 0 1 0 1 1 .. .. || .. .x | 0 0 0 2 | * * * * * * * * * n | 0 0 0 0 0 0 0 1 0 0 1 1 | 0 0 0 1 1 1 -----------------------------+-----------+------------------------+--------------------------+------------ x.-n/d-o. || .. .. | n 0 0 0 | n 0 0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * | 1 1 0 0 0 0 xx ..&#x || .. .. | 2 2 0 0 | 1 2 0 1 0 0 0 0 0 0 | * n * * * * * * * * * * | 1 0 1 0 0 0 .. ox&#x || .. .. | 1 2 0 0 | 0 2 0 0 1 0 0 0 0 0 | * * n * * * * * * * * * | 1 0 0 1 0 0 x. .. || x. .. | 2 0 2 0 | 1 0 2 0 0 0 1 0 0 0 | * * * n * * * * * * * * | 0 1 1 0 0 0 oo-n/d-oo&#x || oo-n/d-oo&#x | 1 1 1 1 | 0 1 1 0 0 1 0 1 0 0 | * * * * 2n * * * * * * * | 0 0 1 1 0 0 .x-n/d-.x || .. .. | 0 2n 0 0 | 0 0 0 n n 0 0 0 0 0 | * * * * * 1 * * * * * * | 1 0 0 0 1 0 .x .. || .x .. | 0 2 0 2 | 0 0 0 1 0 2 0 0 1 0 | * * * * * * n * * * * * | 0 0 1 0 1 0 .. .x || .. .x | 0 2 0 2 | 0 0 0 0 1 2 0 0 0 1 | * * * * * * * n * * * * | 0 0 0 1 1 0 .. .. || x.-n/d-o. | 0 0 n 0 | 0 0 0 0 0 0 n 0 0 0 | * * * * * * * * 1 * * * | 0 1 0 0 0 1 .. .. || xx ..&#x | 0 0 2 2 | 0 0 0 0 0 0 1 2 1 0 | * * * * * * * * * n * * | 0 0 1 0 0 1 .. .. || .. ox&#x | 0 0 1 2 | 0 0 0 0 0 0 0 2 0 1 | * * * * * * * * * * n * | 0 0 0 1 0 1 .. .. || .x-n/d-.x | 0 0 0 2n | 0 0 0 0 0 0 0 0 n n | * * * * * * * * * * * 1 | 0 0 0 0 1 1 -----------------------------+-----------+------------------------+--------------------------+------------ xx-n/d-ox&#x || .. .. ♦ n 2n 0 0 | n 2n 0 n n 0 0 0 0 0 | 1 n n 0 0 1 0 0 0 0 0 0 | 1 * * * * * x.-n/d-o. || x.-n/d-o. ♦ n 0 n 0 | n 0 n 0 0 0 n 0 0 0 | 1 0 0 n 0 0 0 0 1 0 0 0 | * 1 * * * * xx ..&#x || xx ..&#x ♦ 2 2 2 2 | 1 2 2 1 0 2 1 2 1 0 | 0 1 0 1 2 0 1 0 0 1 0 0 | * * n * * * .. ox&#x || .. ox&#x ♦ 1 2 1 2 | 0 2 1 0 1 2 0 2 0 1 | 0 0 1 0 2 0 0 1 0 0 1 0 | * * * n * * .x-n/d-.x || .x-n/d-.x ♦ 0 2n 0 2n | 0 0 0 n n 2n 0 0 n n | 0 0 0 0 0 1 n n 0 0 0 1 | * * * * 1 * .. .. || xx-n/d-ox&#x ♦ 0 0 n 2n | 0 0 0 0 0 0 n 2n n n | 0 0 0 0 0 0 0 0 1 n n 1 | * * * * * 1
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